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Properties

Label 1-431-431.147-r0-0-0
Degree $1$
Conductor $431$
Sign $0.127 - 0.991i$
Analytic cond. $2.00155$
Root an. cond. $2.00155$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.181 + 0.983i)2-s + (−0.694 − 0.719i)3-s + (−0.934 − 0.357i)4-s + (−0.238 + 0.971i)5-s + (0.833 − 0.551i)6-s + (−0.557 − 0.829i)7-s + (0.520 − 0.853i)8-s + (−0.0365 + 0.999i)9-s + (−0.911 − 0.411i)10-s + (0.683 + 0.729i)11-s + (0.391 + 0.920i)12-s + (−0.991 − 0.131i)13-s + (0.917 − 0.397i)14-s + (0.864 − 0.502i)15-s + (0.744 + 0.667i)16-s + (−0.483 + 0.875i)17-s + ⋯
L(s)  = 1  + (−0.181 + 0.983i)2-s + (−0.694 − 0.719i)3-s + (−0.934 − 0.357i)4-s + (−0.238 + 0.971i)5-s + (0.833 − 0.551i)6-s + (−0.557 − 0.829i)7-s + (0.520 − 0.853i)8-s + (−0.0365 + 0.999i)9-s + (−0.911 − 0.411i)10-s + (0.683 + 0.729i)11-s + (0.391 + 0.920i)12-s + (−0.991 − 0.131i)13-s + (0.917 − 0.397i)14-s + (0.864 − 0.502i)15-s + (0.744 + 0.667i)16-s + (−0.483 + 0.875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.127 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.127 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $0.127 - 0.991i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ 0.127 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1753249077 - 0.1541859350i\)
\(L(\frac12)\) \(\approx\) \(0.1753249077 - 0.1541859350i\)
\(L(1)\) \(\approx\) \(0.5033593934 + 0.1362573912i\)
\(L(1)\) \(\approx\) \(0.5033593934 + 0.1362573912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (-0.181 + 0.983i)T \)
3 \( 1 + (-0.694 - 0.719i)T \)
5 \( 1 + (-0.238 + 0.971i)T \)
7 \( 1 + (-0.557 - 0.829i)T \)
11 \( 1 + (0.683 + 0.729i)T \)
13 \( 1 + (-0.991 - 0.131i)T \)
17 \( 1 + (-0.483 + 0.875i)T \)
19 \( 1 + (0.763 + 0.645i)T \)
23 \( 1 + (0.281 - 0.959i)T \)
29 \( 1 + (-0.628 - 0.777i)T \)
31 \( 1 + (-0.350 - 0.936i)T \)
37 \( 1 + (-0.508 + 0.861i)T \)
41 \( 1 + (-0.210 - 0.977i)T \)
43 \( 1 + (-0.911 + 0.411i)T \)
47 \( 1 + (-0.872 - 0.489i)T \)
53 \( 1 + (0.417 - 0.908i)T \)
59 \( 1 + (0.724 - 0.688i)T \)
61 \( 1 + (0.817 - 0.575i)T \)
67 \( 1 + (-0.0948 - 0.995i)T \)
71 \( 1 + (-0.734 + 0.678i)T \)
73 \( 1 + (-0.00730 - 0.999i)T \)
79 \( 1 + (-0.987 - 0.160i)T \)
83 \( 1 + (0.470 - 0.882i)T \)
89 \( 1 + (-0.605 - 0.795i)T \)
97 \( 1 + (0.704 - 0.709i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.21256925500884202214716730543, −23.207509756321146971731327784151, −22.16088474404764180699990389878, −21.87444988187377707118152112188, −21.00857036981657484421277713488, −19.94193875523283530496202019203, −19.49100080902333844609145211999, −18.24548513719420167035880717968, −17.44165453659276013186043130023, −16.517717117965374901492334130032, −15.94514569165870579897169912068, −14.733299806474526115230557187365, −13.4636117790602180522053955313, −12.51656579209226891149551410996, −11.74654141284491951206237979577, −11.32533178114545265639409955883, −9.8990700394563534663286457640, −9.198406379214059836291381077863, −8.78146114679742158375220818648, −7.09753812755752597054461734245, −5.461558265415332398322047926593, −5.02229922336596899733628742651, −3.832598783066053791817123793732, −2.88675556156517489658247959517, −1.22632505606820569021970281583, 0.174089625056651179347372645040, 1.87957955357433460514402572141, 3.65271286558587341045742277192, 4.71136040468921598760114458297, 6.01669935965804460076500666473, 6.79584672025207101855059197649, 7.26260581774205752004827092512, 8.15277276176363625230068399060, 9.81562505322139966205937286823, 10.30584494943182003895055260701, 11.53714005081887198699162581950, 12.62549771578644901297067248836, 13.465664806803436281558850393190, 14.45152375720665095513754046100, 15.10602897516705098100214223634, 16.33311438680026703965542400644, 17.112220087116981419026708095966, 17.616631668871018751448103534881, 18.67544360778987804418913142407, 19.256531881097852457711826068172, 20.11787161725970906517523943790, 22.10216241113731134389032827271, 22.52908886254218949689289664663, 23.0063683075235410893149818873, 24.00178284338776431742484903831

Graph of the $Z$-function along the critical line